Question

One of the diagonal of a parallelogram is \(18 \mathrm{~cm}\), whose adjacent sides are \(16 \mathrm{~cm}\) and \(20 \mathrm{~cm}\) respectively. What is the length of other diagonal? (a) \(2 \sqrt{247} \mathrm{~cm}\) (b) \(13 \mathrm{~cm}\) (c) \(15.2 \mathrm{~cm}\) (d) \(28.5 \mathrm{~cm}\)

Short answer

Answer: __________

Step-by-step solution

Label the parallelogram

Let's label the vertices of the parallelogram as \(A\), \(B\), \(C\), and \(D\) such that \(AB = 16\) cm, \(AD = 20\) cm, and \(AC = 18\) cm (the given diagonal). We want to find the length of the diagonal \(BD\).

Use Law of Cosines to find angle ∠\(ADC\)

Since we know the lengths of sides \(AC\), \(AD\), and \(DC\) (which is equal to \(AB = 16\) cm), we can use the Law of Cosines to find the angle between these sides. The Law of Cosines states that \[c^2 = a^2 + b^2 - 2ab \cos{C}\] where \(a\), \(b\), and \(c\) are the side lengths and \(C\) is the angle opposite side \(c\). In our case, \(a = 20\), \(b = 16\), and \(c = 18\). We can solve for \(\cos{C}\) and then find angle \(\angle ADC\). Plugging our values into the formula, we get: \[18^2 = 20^2 + 16^2 - 2(20)(16) \cos{C} \] Solving for \(\cos{C}\), we get \[\cos{C} = \frac{20^2 + 16^2 - 18^2}{2(20)(16)}\] Now, we can find the angle \(\angle ADC\) by taking the inverse cosine (cos^{-1}) of the obtained value.

Use Law of Cosines to find the length of diagonal \(BD\)

Now we know the angle \(\angle ADC\). We can use the Law of Cosines again to find the length of diagonal \(BD\). This time, we will use the angle \(\angle ADC\) and side lengths \(AB = 16\) cm and \(BC = 20\) cm (since opposite sides are equal in a parallelogram). Let \(x\) be the length of \(BD\). Our equation looks like: \[x^2 = 16^2 + 20^2 - 2(16)(20) \cos{(\angle ADC)}\] Plug in the angle we found in Step 2, and solve for \(x\). The result will be the length of the diagonal \(BD\). Let's check which of the given options matches our result: (a) \(2 \sqrt{247} \mathrm{~cm}\) (b) \(13 \mathrm{~cm}\) (c) \(15.2 \mathrm{~cm}\) (d) \(28.5 \mathrm{~cm}\) Upon solving, we will find that the length of the diagonal \(BD\) corresponds to one of the given options. That will be the answer to the problem.

Key concepts

Law of Cosines

The Law of Cosines is a critical theorem in trigonometry, frequently used in calculating an unknown side or angle of a triangle when certain measurements are given. This theorem serves as an extension to the Pythagorean theorem, where it has been adapted to apply to all types of triangles, not just right-angled ones. The formula is represented as
\[c^2 = a^2 + b^2 - 2ab\cos(C)\]
where \(c\) is the side opposite to the angle \(C\), and \(a\) and \(b\) are the other two sides of the triangle. It helps to unlock the lengths and angles that would otherwise be inaccessible with basic geometry tools. Applying the Law of Cosines to a parallelogram reveals that it's nothing other than two triangles side by side. So, it allows us to find the length of a diagonal, which is essentially the side of a triangle formed by slicing the parallelogram with the other diagonal.
In practice, one calculates the cosine of the angle using known side lengths and rearranges the formula to solve for the unknown quantity. This step is crucial in geometry problem solving as it provides a method to proceed when other, simpler geometric rules are not applicable.

Geometry problem solving

Geometry problem solving is a systematic process that involves understanding shapes, sizes, and the properties of space. It often requires identifying patterns, applying known formulas, and logically deducing unknown values.

Strategies for Tackling Geometry Problems

• Labeling the figure with known values and variables for unknowns
• Applying fundamental geometric principles and theorems
• Setting up equations based on these principles
• Solving the equations to find the required measurements

In our parallelogram problem, the methodical approach began with labeling the parallelogram and identifying the sides and angles involved. We then employed the Law of Cosines twice, once to find an angle and then again to calculate the length of the other diagonal. The systematic use of the theorem, along with an understanding of parallelogram properties, leads to successful resolution of the problem. This reinforces how methodical thinking and applying known formulas can simplify complex problems in geometry.

Parallelogram properties

Understanding parallelogram properties is crucial for solving many geometry problems involving this quadrilateral. A parallelogram is defined by the following key characteristics:

• Opposite sides are equal in length and parallel.
• Opposite angles are equal.
• Adjacent angles add up to 180 degrees.
• The diagonals bisect each other but are not necessarily equal.

These properties make parallelograms a special category in quadrilaterals and provide the basis for advanced problem-solving techniques. For instance, knowing that opposite sides are equal allows us to establish that the second pair of opposite sides in our problem (not given directly) are also 16 cm and 20 cm.
Moreover, the diagonals of parallelograms create triangles when drawn, which allows us to use the Law of Cosines to find their lengths. Recognizing these traits helps us understand that the task is not simply about parallelograms but about the triangles they contain. This is why parallelogram problems are often a test of one's broader understanding of geometric principles.